en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(sqrt(e^{2x)+6)}

Solution

\int \frac{1}{e ^{2 x} + 6} d x

Solution

- \frac{1}{2 6} (ln \frac{e ^{2 x} + 6}{6} + 1 - ln \frac{e ^{2 x} + 6}{6} - 1) + C

Solution steps

\int \frac{1}{e ^{2 x} + 6} d x

Apply u-substitution

: \int \frac{1}{2 e ^{u} + 6} d u

= \int \frac{1}{2 e ^{u} + 6} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{e ^{u} + 6} d u

Apply u-substitution

: \int \frac{1}{v v + 6} d v

= \frac{1}{2} \cdot \int \frac{1}{v v + 6} d v

Apply u-substitution

: \int \frac{2}{w ^{2} - 6} d w

= \frac{1}{2} \cdot \int \frac{2}{w ^{2} - 6} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{w ^{2} - 6} d w

Apply Integral Substitution

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{6 ( t ^{2} - 1 )} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \frac{1}{6} \cdot \int \frac{1}{t ^{2} - 1} d t

Factor

t^{2} - 1 : - (- t^{2} + 1)

= \frac{1}{2} \cdot 2 \cdot \frac{1}{6} \cdot \int \frac{1}{- ( - t ^{2} + 1 )} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \frac{1}{6} (- \int \frac{1}{- t ^{2} + 1} d t)

Use the common integral:

\int \frac{1}{- t ^{2} + 1} d t = \frac{ln ∣ t + 1 ∣}{2} - \frac{ln ∣ t - 1 ∣}{2}

= \frac{1}{2} \cdot 2 \cdot \frac{1}{6} (- (\frac{ln ∣ t + 1 ∣}{2} - \frac{ln ∣ t - 1 ∣}{2}))

Substitute back

= \frac{1}{2} \cdot 2 \cdot \frac{1}{6} - \frac{ln \frac{e ^{2 x} + 6}{6} + 1}{2} - \frac{ln \frac{e ^{2 x} + 6}{6} - 1}{2}

Simplify

\frac{1}{2} \cdot 2 \cdot \frac{1}{6} - \frac{ln \frac{e ^{2 x} + 6}{6} + 1}{2} - \frac{ln \frac{e ^{2 x} + 6}{6} - 1}{2} : - \frac{1}{2 6} (ln \frac{e ^{2 x} + 6}{6} + 1 - ln \frac{e ^{2 x} + 6}{6} - 1)

= - \frac{1}{2 6} (ln \frac{e ^{2 x} + 6}{6} + 1 - ln \frac{e ^{2 x} + 6}{6} - 1)

Add a constant to the solution

= - \frac{1}{2 6} (ln \frac{e ^{2 x} + 6}{6} + 1 - ln \frac{e ^{2 x} + 6}{6} - 1) + C

Popular Examples

integral of sqrt(sin(x)-sin^3(x))

\int sin (x) - sin^{3} (x) d x

integral of ln(3+x)

\int ln (3 + x) d x

integral of-10x^2e^{x^3}

\int - 10 x^{2} e^{x^{3}} d x

integral of t/(sqrt(5t+3))

\int \frac{t}{5 t + 3} d t

integral of 1/((x-1)(x+1)^3)

\int \frac{1}{( x - 1 ) ( x + 1 ) ^{3}} d x